The Vlasov-Poisson system with infinite mass and energy

This paper deals with solutions to the Vlasov-Poisson system with an infinite mass. The solution to the Poisson equation cannot be defined directly because the macroscopic density is constant at infinity. To solve this problem, we decompose the solution to the kinetic equation into a homogeneous function and a perturbation. We are then able to prove an existence result in short time for weak solutions to the equation for the perturbation, even though there are no a priori estimates by lack of positivity.